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Lectures 10 and 11

The Risk and Term Structure of Interest Rates

FIGURE 1 Long-Term Bond Yields, 1919–2008

Sources: Board of Governors of the Federal Reserve System, Banking and Monetary Statistics, 1941–1970; Federal Reserve: www.federalreserve.gov/releases/h15/data.htm.

Risk Structure of Interest Rates

Bonds with the same maturity have different interest rates due to:Default riskLiquidity Tax considerations

Risk Structure of Interest Rates

Default risk: probability that the issuer of the bond is unable or unwilling to make interest payments or pay off the face valueU.S. Treasury bonds are considered default

free (government can raise taxes). Risk premium: the spread between the interest

rates on bonds with default risk and the interest rates on (same maturity) Treasury bonds

FIGURE 2 Response to an Increase in Default Risk on Corporate Bonds

Table 1 Bond Ratings by Moody’s, Standard and Poor’s, and Fitch

Risk Structure of Interest Rates

Liquidity: the relative ease with which an asset can be converted into cashCost of selling a bond

Number of buyers/sellers in a bond market

Income tax considerations Interest payments on municipal bonds are

exempt from federal income taxes.

FIGURE 3 Interest Rates on Municipal and Treasury Bonds

Term Structure of Interest Rates

Bonds with identical risk, liquidity, and tax characteristics may have different interest rates because the time remaining to maturity is different

Term Structure of Interest Rates

Yield curve: a plot of the yield on bonds with differing terms to maturity but the same risk, liquidity and tax considerationsUpward-sloping: long-term rates are above

short-term rates

Flat: short- and long-term rates are the same

Inverted: long-term rates are below short-term rates

Facts Theory of the Term Structure of Interest Rates Must Explain

1. Interest rates on bonds of different maturities move together over time

2. When short-term interest rates are low, yield curves are more likely to have an upward slope; when short-term rates are high, yield curves are more likely to slope downward and be inverted

3. Yield curves almost always slope upward

Three Theories to Explain the Three Facts

1. Expectations theory explains the first two facts but not the third

2. Segmented markets theory explains fact three but not the first two

3. Liquidity premium theory combines the two theories to explain all three facts

FIGURE 4 Movements over Time of Interest Rates on U.S. Government Bonds with Different Maturities

Sources: Federal Reserve: www.federalreserve.gov/releases/h15/data.htm.

Expectations Theory

The interest rate on a long-term bond will equal an average of the short-term interest rates that people expect to occur over the life of the long-term bond

Buyers of bonds do not prefer bonds of one maturity over another; they will not hold any quantity of a bond if its expected return is less than that of another bond with a different maturity

Bond holders consider bonds with different maturities to be perfect substitutes

Expectations Theory: Example

Let the current rate on one-year bond be 6%.

You expect the interest rate on a one-year bond to be 8% next year.

Then the expected return for buying two one-year bonds averages (6% + 8%)/2 = 7%.

The interest rate on a two-year bond must be 7% for you to be willing to purchase it.

Expectations Theory

1

2

For an investment of $1

= today's interest rate on a one-period bond

= interest rate on a one-period bond expected for next period

= today's interest rate on the two-period bond

t

et

t

i

i

i

Expectations Theory (cont’d)

2 2

22 2

22 2

22

Expected return over the two periods from investing $1 in the

two-period bond and holding it for the two periods

(1 + )(1 + ) 1

1 2 ( ) 1

2 ( )

Since ( ) is very small

the expected re

t t

t t

t t

t

i i

i i

i i

i

2

turn for holding the two-period bond for two periods is

2 ti

Expectations Theory (cont’d)

1

1 1

1 1

1

1

If two one-period bonds are bought with the $1 investment

(1 )(1 ) 1

1 ( ) 1

( )

( ) is extremely small

Simplifying we get

et t

e et t t t

e et t t t

et t

et t

i i

i i i i

i i i i

i i

i i

Expectations Theory (cont’d)

2 1

12

Both bonds will be held only if the expected returns are equal

2

2The two-period rate must equal the average of the two one-period rates

For bonds with longer maturities

et t t

et t

t

t tnt

i i i

i ii

i ii

1 2 ( 1)...

The -period interest rate equals the average of the one-period

interest rates expected to occur over the -period life of the bond

e e et t ni i

nn

n

Expectations Theory

Explains why the term structure of interest rates changes at different times

Explains why interest rates on bonds with different maturities move together over time (fact 1)

Explains why yield curves tend to slope up when short-term rates are low and slope down when short-term rates are high (fact 2)

Cannot explain why yield curves usually slope upward (fact 3)

Segmented Markets Theory

Bonds of different maturities are not substitutes at all

The interest rate for each bond with a different maturity is determined by the demand for and supply of that bond

Investors have preferences for bonds of one maturity over another

If investors generally prefer bonds with shorter maturities that have less interest-rate risk, then this explains why yield curves usually slope upward (fact 3)

Liquidity Premium & Preferred Habitat Theories

The interest rate on a long-term bond will equal an average of short-term interest rates expected to occur over the life of the long-term bond plus a liquidity premium that responds to supply and demand conditions for that bond

Bonds of different maturities are partial (not perfect) substitutes

Liquidity Premium Theory

int

it i

t1e i

t2e ... i

t(n 1)e

n l

nt

where lnt

is the liquidity premium for the n-period bond at time t

lnt

is always positive

Rises with the term to maturity

Preferred Habitat Theory

Investors have a preference for bonds of one maturity over another

They will be willing to buy bonds of different maturities only if they earn a somewhat higher expected return

Investors are likely to prefer short-term bonds over longer-term bonds

FIGURE 5 The Relationship Between the Liquidity Premium (Preferred Habitat) and Expectations Theory

Liquidity Premium and Preferred Habitat Theories

Interest rates on different maturity bonds move together over time; explained by the first term in the equation

Yield curves tend to slope upward when short-term rates are low and to be inverted when short-term rates are high; explained by the liquidity premium term in the first case and by a low expected average in the second case

Yield curves typically slope upward; explained by a larger liquidity premium as the term to maturity lengthens

FIGURE 6 Yield Curves and the Market’s Expectations of Future Short-Term Interest Rates According to the Liquidity Premium (Preferred Habitat) Theory

FIGURE 7 Yield Curves for U.S. Government Bonds

Sources: Federal Reserve Bank of St. Louis; U.S. Financial Data, various issues; Wall Street Journal, various dates.

Application: The Subprime Collapse and the Baa-Treasury Spread

Corporate Bond Risk Premium and Flight to Quality

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Corporate bonds, monthly data Aaa-RateCorporate bonds, monthly data Baa-Rate10-year maturity Treasury bonds, monthly data